Graph algebras and extremal graph theory

1. Graph algebras and extremal graph theory LászlóLovász

3. Three questions in 2012 Mike Freedman: Whichgraphparameters arepartitionfunctions? JenniferChayes: What is the limit of a randomly growinggraphsequence (like the internet)? Vera Sós: Howtocharacterizegeneralized quasirandomgraphs?

4. Three questions in 2012 Mike Freedman: Whichgraphparameters arepartitionfunctions? JenniferChayes: What is the limit of a randomly growinggraphsequence (like the internet)? Vera Sós: Howtocharacterizegeneralized quasirandomgraphs?

5. Which parameters are homomorphism functions? Graph parameter: isomorphism-invariant function on finite graphs Homomorphism: adjacency-preserving map Partition functions of vertex coloring models

6. Which parameters are homomorphism functions? Probability that random map V(F)V(G)is a hom Weighted version:

7. Which parameters are homomorphism functions? 1 2 k-labeled graph: knodes labeled 1,...,k, any number of unlabeled nodes

8. Connection matrices ... M(f, k) k=2: ...

9. Which parameters are homomorphism functions? Difficult direction: Easy but useful direction: f = hom(.,H) for some weighted graph H on q nodes  (k) M(f,k) is positive semidefinite, rank qk Freedman - L - Schrijver

10. Related (?) characterizations dual problem, categories L-Schrijver tensor algebras Schrijver f=t(.,h) for some ``edge coloring model’’ W Szegedy f=hom(.,H) for some edge-weighted graph H Schrijver f=hom(.,G) for some simple graph G L-Schrijver f=t(.,W) for some ``graphon’’ (graph limit) W L-Szegedy

11. Limit objects (graphons)

12. Graphs  Graphons 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 G AG WG

13. Limit objects (graphons) t(F,WG)=t(F,G) (G1,G2,…) convergent: simple Ft(F,Gn) converges Borgs-Chayes-L-Sós-Vesztergombi

14. Limit objects For every convergent graph sequence (Gn) there is a graphon W such thatGnW. For every graphon Wthere is a graph sequence (Gn)such that GnW. L-Szegedy W is essentially unique (up to measure-preserving transformation). Borgs-Chayes-L f=t(.,W) for some graphon W  f is multiplicative, reflection positive, and f(K1)=1. L-Szegedy

16. Computing with graphs k-labeled quantum graph: 1 finite formal sum of k-labeled graphs 2 infinite dimensional linear spaces infinite formal sum with G |xG| Gk = {k-labeled quantum graphs} Lk= {k-labeled quantum graphs (infinite sums)}

17. Computing with graphs ... is a commutative algebra with unit element Define products: G1,G2: k-labeled graphs G1G2 = G1G2, labeled nodes identified

18. Computing with graphs extend linearly Inner product: f:graph parameter

19. Computing with graphs f is reflection positive

20. Computing with graphs Factor out the kernel:

21. Computing with graphs Assume that rk(M(f,k))=qk. In particular, for k=1:

22. Computing with graphs Assume that rk(M(f,k))=qk. pipj form a basis of G2.

24. Some old and new results from extremal graph theory Extremal: Turán’s Theorem (special case proved by Mantel): G contains no triangles  #edgesn2/4 Theorem (Goodman):t(K3,G)  (2t(K2,G) -1) t(K2,G)

25. Some old and new results from extremal graph theory Theorem (Erdős): G contains no 4-cycles  #edgesn3/2/2 (Extremal: conjugacy graph of finite projective planes) t(C4,G)  t(K2,G)4

26. Connection matrices k=2: t( ,G)  t( ,G)2  t( ,G)4

27. Which inequalities between densities are valid? IfvalidforlargeG, thenvalidforall Undecidable… Hatami-Norine …but decidable with an arbitrarily small error. L-Szegedy

28. Computing with graphs -2 + Turán: Kruskal-Katona: - - Blakley-Roy: Writex ≥ 0 if hom(x,G) ≥ 0for every graph G.

29. Computing with graphs 2 - - + = - - + 2 - +2 = 2 -4 2 2 - - - + = - - +2 + -4 +2 +2  - 2 + - 2 + ≥ 0 Goodman’s Theorem t( ,G) – 2t( ,G) + t( ,G) ≥ 0

30. Positivstellensatz for graphs? If a quantum graph x is a sum of squares (ignoring labels and isolated nodes), then x ≥ 0. Question: Suppose thatx ≥ 0. Does it follow that No! Hatami-Norine

31. A weak Positivstellensatz for graphs Let x be a quantum graph. Then x  0  L-Szegedy

32. An exact Positivstellensatz for graphs F*:graphs partially labeled by different nonnegative integers L: functions f: F* such that F |f(F)|<.

33. An exact Positivstellensatz for graphs x 0 

34. Proof of the weak Positivstellensatz (sketch2) the optimum of a semidefinite program is 0: minimize subject to M(f,k)positive semidefinite for all k f(K1)=1 f(GK1)=f(G) Apply Duality Theorem of semidefinite programming

35. Pentagon density in triangle-free graphs

36. Pentagon density in triangle-free graphs Conjecture: Erdős 1984 Proof: Hatami-Hladky-Král-Norine-Razborov 2011

37. Pentagon density in triangle-free graphs 3-labeled quantum graphs positive semidefinite

39. Many more great theorems, Lex!